This MO answer cites the Goodman-Wallach book to affirm that: $$\mathrm{Sym}^k\left(\mathbb{R}^n\right)=\mathcal{H}^k\oplus q\mathcal{H}^{k-2}\oplus q^2\mathcal{H}^{k-4}\oplus\cdots$$ with $\mathrm{Sym}^k\left(\mathbb{R}^n\right)$ being the subspace of vectors left invariant by permutation of their tensor factors, $\mathcal{H}^p$ being the space of harmonic polynomials with $n$ variables of degree $p$ killed by the Laplacian: $$\mathcal{H}^p=\left\{P\in\mathcal{P}^p\middle|\sum_{i=1}^n\frac{\partial P}{\partial X_i^2}=0\right\}$$ with $\mathcal{P}^p$ being the space of homogeneoous polynomials with $n$ variables of degree $p$: $$\mathcal{P}^p=\mathrm{span}\left\{X_1^{p_1}\cdots X_n^{p_n}\middle|\sum_{i=1}^np_i=p\right\}$$ and with $q=\sum\limits_{i=1}^nX_i^2$.
The answer then affirms that all spaces $q^l\mathcal{H}^p$ are irreducible representations of the group action $O\in\mathcal{O}(n)\mapsto O^{\otimes (l+p)}$ for $n\geqslant2$, with $\mathcal{O}(n)$ being the orthogonal group and $\otimes$ being the tensor product. Is there a reference for this result? Or am I missing something simple enough for the proof not to be provided?
